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Register a new userOn the emergence of orientational order in folded proteins with implications for allostery
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The beautiful structures of single and multi-domain proteins are clearly ordered in some fashion but cannot be readily classified using group theory methods that are successfully used to describe periodic crystals. For this reason, protein structures are considered to be aperiodic, and may have evolved this way for functional purposes, especially in instances that require a combination of softness and rigidity within the same molecule. By analyzing the solved protein structures, we show that orientational symmetry is broken in the aperiodic arrangement of the secondary structural elements (SSEs), which we deduce by calculating the nematic order parameter, $P_{2}$. We find that the folded structures are nematic droplets with a broad distribution of $P_{2}$. We argue that non-zero values of $P_{2}$, leads to an arrangement of the SSEs that can resist mechanical forces, which is a requirement for allosteric proteins. Such proteins, which resist mechanical forces in some regions while being flexible in others, transmit signals from one region of the protein to another (action at a distance) in response to binding of ligands (oxygen, ATP or other small molecules).
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Chiral heteropolymers such as larger globular proteins can simultaneously support multiple length scales. The interplay between different scales brings about conformational diversity, and governs the structure of the energy landscape. Multiple scales produces also complex dynamics, which in the case of proteins sustains live matter. However, thus far no clear understanding exist, how to distinguish the various scales that determine the structure and dynamics of a complex protein. Here we propose a systematic method to identify the scales in chiral heteropolymers such as a protein. For this we introduce a novel order parameter, that not only reveals the scales but also probes the phase structure. In particular, we argue that a chiral heteropolymer can simultaneously display traits of several different phases, contingent on the length scale at which it is scrutinized. Our approach builds on a variant of Kadanoffs block-spin transformation that we employ to coarse grain piecewise linear chains such as the C$alpha$ backbone of a protein. We derive analytically and then verify numerically a number of properties that the order parameter can display. We demonstrate how, in the case of crystallographic protein structures in Protein Data Bank, the order parameter reveals the presence of different length scales, and we propose that a relation must exist between the scales, phases, and the complexity of folding pathways.
Previous studies of the flexibilities of ancestral proteins suggests that proteins evolve their function by altering their native state ensemble. Here we propose a more direct method of visualizing this by measuring the changes in the vibrational density of states (VDOS) of proteins as they evolve. Through analysis of VDOS profiles of ancestral and extant proteins we observe that $beta$-lactamase and thioredoxins evolve by altering their density of states in the terahertz region. Particularly, the shift in VDOS profiles between ancestral and extant proteins suggests that nature utilize dynamic allostery for functional evolution. Moreover, we also show that VDOS profile of individual position can be used to describe the flexibility changes, particularly those without any amino acid substitution.
We introduce the spatial correlation function $C_Q(r)$ and temporal autocorrelation function $C_Q(t)$ of the local tetrahedral order parameter $Qequiv Q(r,t)$. Using computer simulations of the TIP5P model of water, we investigate $C_Q(r)$ in a broad region of the phase diagram. First we show that $C_Q(r)$ displays anticorrelation at $rapprox 0.32$nm at high temperatures $T>T_Wapprox 250$ K, which changes to positive correlation below the Widom line $T_W$. Further we find that at low temperatures $C_Q(t)$ exhibits a two-step temporal decay similar to the self intermediate scattering function, and that the corresponding correlation time $tau_Q$ displays a dynamic crossover from non-Arrhenius behavior for $T>T_W$ to Arrhenius behavior for $T<T_W$. Finally, we define an orientational entropy $S_Q$ associated with the {it local} orientational order of water molecules, and show that $tau_Q$ can be extracted from $S_Q$ using an analog of the Adam-Gibbs relation.
Motivated by experiments on colloidal membranes composed of chiral rod-like viruses, we use Monte Carlo methods to determine the phase diagram for the liquid crystalline order of the rods and the membrane shape. We generalize the Lebwohl-Lasher model for a nematic with a chiral coupling to a curved surface with edge tension and a resistance to bending, and include an energy cost for tilting of the rods relative to the local membrane normal. The membrane is represented by a triangular mesh of hard beads joined by bonds, where each bead is decorated by a director. The beads can move, the bonds can reconnect and the directors can rotate at each Monte Carlo step. When the cost of tilt is small, the membrane tends to be flat, with the rods only twisting near the edge for low chiral coupling, and remaining parallel to the normal in the interior of the membrane. At high chiral coupling, the rods twist everywhere, forming a cholesteric state. When the cost of tilt is large, the emergence of the cholesteric state at high values of the chiral coupling is accompanied by the bending of the membrane into a saddle shape. Increasing the edge tension tends to flatten the membrane. These results illustrate the geometric frustration arising from the inability of a surface normal to have twist.
The free energy of globular protein chain is considered to be a functional defined on smooth curves in three dimensional Euclidean space. From the requirement of geometrical invariance, together with basic facts on conformation of helical proteins and dynamical characteristics of the protein chains, we are able to determine, in a unique way, the exact form of the free energy functional. Namely, the free energy density should be a linear function of the curvature of curves on which the free energy functional is defined. This model can be used, for example, in Monte Carlo simulations of exhaustive searching the native stable state of the protein chain.
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